Mixed finite difference of a polynomial
= Mixed finite difference of a polynomial
If $p$ has degree $n$ and leading coefficient $a_n$, then for nonzero steps $r_1,\ldots,r_n$,
$$
D_{r_1}\cdots D_{r_n}p=n!a_n\prod_{i=1}^nr_i.
$$
Consequently a degree-$n$ polynomial cannot be the sum of $n$ functions each periodic with some nonzero period.